Theorem sibfima 34128

Description: Any preimage of a singleton by a simple function is measurable. (Contributed by Thierry Arnoux, 19-Feb-2018.)

Hypotheses

Ref	Expression
sitgval.b	⊢ 𝐵 = (Base‘𝑊)
sitgval.j	⊢ 𝐽 = (TopOpen‘𝑊)
sitgval.s	⊢ 𝑆 = (sigaGen‘𝐽)
sitgval.0	⊢ 0 = (0g‘𝑊)
sitgval.x	⊢ · = ( ·𝑠 ‘𝑊)
sitgval.h	⊢ 𝐻 = (ℝHom‘(Scalar‘𝑊))
sitgval.1	⊢ (𝜑 → 𝑊 ∈ 𝑉)
sitgval.2	⊢ (𝜑 → 𝑀 ∈ ∪ ran measures)
sibfmbl.1	⊢ (𝜑 → 𝐹 ∈ dom (𝑊sitg𝑀))

Assertion

Ref	Expression
sibfima	⊢ ((𝜑 ∧ 𝐴 ∈ (ran 𝐹 ∖ { 0 })) → (𝑀‘(◡𝐹 “ {𝐴})) ∈ (0[,)+∞))

Proof of Theorem sibfima

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		sibfmbl.1	. . . 4 ⊢ (𝜑 → 𝐹 ∈ dom (𝑊sitg𝑀))
2		sitgval.b	. . . . 5 ⊢ 𝐵 = (Base‘𝑊)
3		sitgval.j	. . . . 5 ⊢ 𝐽 = (TopOpen‘𝑊)
4		sitgval.s	. . . . 5 ⊢ 𝑆 = (sigaGen‘𝐽)
5		sitgval.0	. . . . 5 ⊢ 0 = (0g‘𝑊)
6		sitgval.x	. . . . 5 ⊢ · = ( ·𝑠 ‘𝑊)
7		sitgval.h	. . . . 5 ⊢ 𝐻 = (ℝHom‘(Scalar‘𝑊))
8		sitgval.1	. . . . 5 ⊢ (𝜑 → 𝑊 ∈ 𝑉)
9		sitgval.2	. . . . 5 ⊢ (𝜑 → 𝑀 ∈ ∪ ran measures)
10	2, 3, 4, 5, 6, 7, 8, 9	issibf 34123	. . . 4 ⊢ (𝜑 → (𝐹 ∈ dom (𝑊sitg𝑀) ↔ (𝐹 ∈ (dom 𝑀MblFnM𝑆) ∧ ran 𝐹 ∈ Fin ∧ ∀𝑥 ∈ (ran 𝐹 ∖ { 0 })(𝑀‘(◡𝐹 “ {𝑥})) ∈ (0[,)+∞))))
11	1, 10	mpbid 231	. . 3 ⊢ (𝜑 → (𝐹 ∈ (dom 𝑀MblFnM𝑆) ∧ ran 𝐹 ∈ Fin ∧ ∀𝑥 ∈ (ran 𝐹 ∖ { 0 })(𝑀‘(◡𝐹 “ {𝑥})) ∈ (0[,)+∞)))
12	11	simp3d 1141	. 2 ⊢ (𝜑 → ∀𝑥 ∈ (ran 𝐹 ∖ { 0 })(𝑀‘(◡𝐹 “ {𝑥})) ∈ (0[,)+∞))
13		sneq 4642	. . . . . 6 ⊢ (𝑥 = 𝐴 → {𝑥} = {𝐴})
14	13	imaeq2d 6068	. . . . 5 ⊢ (𝑥 = 𝐴 → (◡𝐹 “ {𝑥}) = (◡𝐹 “ {𝐴}))
15	14	fveq2d 6904	. . . 4 ⊢ (𝑥 = 𝐴 → (𝑀‘(◡𝐹 “ {𝑥})) = (𝑀‘(◡𝐹 “ {𝐴})))
16	15	eleq1d 2810	. . 3 ⊢ (𝑥 = 𝐴 → ((𝑀‘(◡𝐹 “ {𝑥})) ∈ (0[,)+∞) ↔ (𝑀‘(◡𝐹 “ {𝐴})) ∈ (0[,)+∞)))
17	16	rspcv 3603	. 2 ⊢ (𝐴 ∈ (ran 𝐹 ∖ { 0 }) → (∀𝑥 ∈ (ran 𝐹 ∖ { 0 })(𝑀‘(◡𝐹 “ {𝑥})) ∈ (0[,)+∞) → (𝑀‘(◡𝐹 “ {𝐴})) ∈ (0[,)+∞)))
18	12, 17	mpan9 505	1 ⊢ ((𝜑 ∧ 𝐴 ∈ (ran 𝐹 ∖ { 0 })) → (𝑀‘(◡𝐹 “ {𝐴})) ∈ (0[,)+∞))

Syntax hints:   → wi 4   ∧ wa 394   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  ∀wral 3050   ∖ cdif 3943  {csn 4632  ∪ cuni 4912  ^◡ccnv 5680  dom cdm 5681  ran crn 5682   “ cima 5684  ‘cfv 6553  (class class class)co 7423  Fincfn 8973  0cc0 11154  +∞cpnf 11291  [,)cico 13375  Basecbs 17208  Scalarcsca 17264   ·_𝑠 cvsca 17265  TopOpenctopn 17431  0_gc0g 17449  ℝHomcrrh 33764  sigaGencsigagen 33927  measurescmeas 33984  MblFnMcmbfm 34038  sitgcsitg 34119

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5289  ax-sep 5303  ax-nul 5310  ax-pr 5432

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3776  df-csb 3892  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-nul 4325  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5579  df-xp 5687  df-rel 5688  df-cnv 5689  df-co 5690  df-dm 5691  df-rn 5692  df-res 5693  df-ima 5694  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-ov 7426  df-oprab 7427  df-mpo 7428  df-sitg 34120

This theorem is referenced by:  sibfinima  34129  sitgfval  34131  sitgclg  34132